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In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.

As cited in [

So far, a lot of works published on fractional order linear/non-linear differential equations but there are still works have to be done. In this work, we aim to extend the Hermite Collocation method (HCM) for obtaining solution to a system of fractional order differential equations with variable coefficients and specified initial conditions. The technique constructs an analytical solution of the form of a truncated Hermite series with unknown coefficients. The orthogonal Hermite polynomials have the importance in the theory of light fluctuations and quantum states and, in particular, some problems of coastal hydrodynamics and meteorology [

This paper is organized as follows. Section 2 involves some basic definitions and properties of fractional calculus. In Section 3, the theory and definitions of Hermite collocation method and the construction of this method for fractional order systems are presented. In Section 4, the matrix relations for initial conditions are defined and the Section 5 deals with the error estimate for the method. Section 6 involves some illustrative examples. Finally, the last section concludes with some remarks based on the reported research.

We first recall the following known definitions and preliminary facts of fractional derivatives and integrals which are used throughout this paper.

Definition 2.1. ( [

where

Definition 2.2. ( [

where

Alternatively, we recall the following definition of Caputo ( [

Definition 2.3. The Caputo fractional derivative of order

Some properties of Caputo derivative are given as follows:

1) ( [

2) ( [

3) ( [

4) Let

In this section, we will consider the following system of fractional order differential equations (FDEs) with variable coefficients,

where

In Equation (5),

where

where

Now, we need to define the Caputo derivatives of

where

If N is an odd number:

if N is even then,

Hence, we have

and letting,

and then, substitution of Equation (11) into Equation (6) yields,

Now, the nath order Caputo derivative of Equation (12) is written as

or equivalently:

Here, the matrix B is defined as follows (

Hence, if we substitute Equation (14) into Equation (13) we have:

Therefore, the matrices in Equation (15), for

where the each submatrix,

where

where

Apart from this, arranging Equation (16) for each collocation points then, we can write explicitly as,

Therefore, the matrix form is equivalent to

where

and each submatrix in

Consequently, now we denote Equation (7) of the form:

Then, by writing Equation (18) in Equation (19), the matrix form of the system of FDEs is written by

Moreover, denoting

hence, the system of FDEs is simply shown by

Now, Equation (21) constructs an algebraic system. To obtain the solution of the above system, the augmented matrix is written as follows:

Solving the above system, as a result, we obtain the desired Hermite coefficients in the truncated Hermite series. Hence, writing

In generally, we look for the solution of the system of FDEs under specified conditions. However, preceding calculations do not involve these conditions. Therefore, we need to incorporate these conditions into the work. Then, we have to establish the new form of Equation (22) which involves initial conditions, Equation (5). Now, we start by writing Equation (5) explicitly for each

Hence, by using the above relations, we obtain t-conditions for each unknown,

Therefore, the conditions in matrix notation fulfils,

where

and for

Now writing Equation (16) into Equation (23) for

Now, calling U as,

then, the Hermite polynomial coefficients matrix which corresponds to the given initial conditions (Equation (5)), can be written as

In Equation (25), U involves kt rows and

Hence, the system of algebraic equations of which unknowns are the hermite polynomial coefficients are shown by

Theorem 1. If

By the above theorem, the matrix of Hermite coefficients, A is uniquely determined by Equation (27). Finally, substitution of these coefficients into the truncated Hermite series gives the desired solution of the form:

The truncated Hermite series, Equation (29), is the approximate solution of Equation (4) with the given initial conditions, (Equation (5)). Since this solution should approximately satisfy the Equation (4) hence, the residuals

give the error at the particular points

The technique which we have developed to solve fractional order systems is quite feasible and accurate. To show the accuracy of the method the following system of FDEs with variable coefficients are solved. All the numerical calculations have been performed by using MatlabR2007b.

Example 6.1. We first consider the problem, which is mentioned in [

with given initial conditions,

Now, we will look for a solution to the system of FDEs in terms of Hermite polynomials of the form;

Here, we will take into consideration:

By using Equations ((6) and (20)) then, the matrix form of the system, Equation (30), is written by

where

For

Hence, we have

Therefore, evaluating Equation (34), we obtain W as,

then, the augmented matrix for the system,

where the matrix

by defining,

Hence, we have,

Then, by substituting the related matrices into Equation (37), the augmented matrix

Moreover, deleting the last two rows of Equation (36) and replacing the matrix in Equation (38), we have

Since

In conclusion, writing these coefficients into:

we obtain the solutions of the system of FDEs as follows

Example 6.2. In [

where they considered:

ordinary system by Bessel Polynomial Collocation method (BCM) with the assumptions:

Now, we will solve the fractional form of Equation (44), which is defined as in Equation (44) by HCM method. We consider here the case:

The fundamental matrix form of the system of Equation (44) is obtained from Equations ((6) and (20)) such as follows,

which is equivalent to:

Then, by performing the calculations, we obtain the following matrices:

then, the agumented matrix

Since,

the solution of the system, Equation (44), is obtained as follows;

The basic goal of this work is to employ HCM method to obtain solution for a system of fractional order differential equations. These types of systems with variable coefficients are usually difficult to solve analytically. However, the presented method provides considerable simplifications in the solution. The coefficients of truncated Hermite series can be evaluated easily by the help of any symbolic computer packages. The obtained results demonstrate the reliability of the algorithm and give us a wider applicability to fractional higher order systems.

Pirim, N.A. and Ayaz, F. (2016) A New Technique for Solving Fractional Order Systems: Hermite Col- location Method. Applied Mathematics, 7, 2307-2323. http://dx.doi.org/10.4236/am.2016.718182